miscellanea-python.ipynb

Kernel: Unknown Kernel

Task 1

This homework is an activity intended to solve some basic problems in python in order to know its basic capabilities.

Due to: May 3

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Miscellanea

Solve the next exercises:

1. Write a program that sort a list of N numbers.

#! /usr/bin/python
import numpy as np

N = 5
#The set is generated randomly
X = np.random.random(N)
print "Original set:", X
print "Sorted set:", np.sort(X)

Original set: [ 0.60423365  0.06462303  0.01382518  0.58877186  0.63548625]
Sorted set: [ 0.01382518  0.06462303  0.58877186  0.60423365  0.63548625]

2. Find all the elements of a list of N numbers that are greater than the mean value of them. Then, create a new list with those numbers.

#! /usr/bin/python
import numpy as np

N = 10
#The set is generated randomly
X = 100*np.random.random(N)
Y = X[ X>=X.mean() ]
print "Original set", X
print "Numbers greater than %1.2f:"%(X.mean()), Y

Original set [  0.88939513  17.434237    58.54453737   3.66019105   5.22876937
  26.54719482  76.98181862  75.8326774   57.25619751  60.1402436 ]
Numbers greater than 38.25: [ 58.54453737  76.98181862  75.8326774   57.25619751  60.1402436 ]

3. Write a program that gives the name of students who passed a course (grade$\geq 3.0\ \$ over $5.0\ \$).

#! /usr/bin/python
import numpy as np

#All students
grades = {"Juanito":2.5, "Pepito":4.8, "Carlitos":3.0, "Maria":3.9, "Pepita":4.6, "Anita":1.9, "Bertulfo":0.9}

#Students that approved
winners = np.array(grades.keys())[ np.array(grades.values())>=3.0 ]
print "The next students approved:", winners

The next students approved: ['Pepito' 'Carlitos' 'Pepita' 'Maria']

4. Write a program that evaluates the next expression for given values of $x$, $a$ and $b$:

$$\frac{\sin(x) + \cos(x)}{2}(a+b)$$

you may find useful the library math for the functions $\sin(x)\ \$ and $\cos(x)$. Import it as import math

5. Find all the prime numbers between 1 and 100.

#! /usr/bin/python
import numpy, math

#Prime Checker
def primes(upto):
    primes = numpy.arange(3,upto+1,2)
    isprime = numpy.ones((upto-1)/2,dtype=bool)
    for factor in primes[:int(math.sqrt(upto))]:
        if isprime[(factor-2)/2]: isprime[(factor*3-2)/2::factor]=0
    return numpy.insert(primes[isprime],0,2)

print "The prime numbers between 1 and 100 are:", primes(100)

The prime numbers between 1 and 100 are: [ 2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97]

6. Write a program that calculates the factorial of a given number.

#! /usr/bin/python
N = 5
#Factorial function
def factorial(N):
    if N==1:
        return 1
    else:
        return N*factorial(N-1)
    
print "The factorial of %d is %d"%(N, factorial(N))

The factorial of 5 is 120

7. Write a program that evaluates the $\cos(x)\ \$ function by using the Taylor expansion using $2$, $5$ and $10$ terms.

$$\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n!)}x^{2n}\ \ \$$

#! /usr/bin/python
from __future__ import division
import math

#Number of terms
N = [2,5,10]
x = 1.5

#Taylor expansion of cos(x)
def taylor_cos( x, N ):
    term = lambda x,n: (-1)**n/(math.factorial(2*n))*x**(2*n)
    return np.sum( [term(x,n) for n in xrange(0,N)] )

for n in N:
    print "Taylor expansion of cos(%1.2f) for %d terms: %1.6f"%( x, n, taylor_cos(x,n) )
print "Exact value of cos(%1.2f): %1.6f"%( x, math.cos(x) )

Taylor expansion of cos(1.50) for 2 terms: -0.125000
Taylor expansion of cos(1.50) for 5 terms: 0.070753
Taylor expansion of cos(1.50) for 10 terms: 0.070737
Exact value of cos(1.50): 0.070737

8. Write a program that calculates $\pi$ by using the next approximations:

$$\frac{2}{\pi} = \frac{\sqrt{2}}{2}\times \frac{\sqrt{2+\sqrt{2}}}{2}\times \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\times \cdots$$

and

$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots$$

Which approximation takes less terms for achieving a good accuracy?